In [1]}, [2]}, [3]} (see also [4]}), Carrollian geometry is described in the metric formulation. In this formulation in [3]}, Carroll-Weyl transformations are defined such that the metric data describing the temporal part of the geometry scales differently than the non-degenerate metric on the spatial hypersurface, the difference in scaling being governed by a real number \(z\) . For \(z = \frac{2}{N}\) , for integer \(N\)  the conformal isometries of the Carrollian spacetime form the conformal Carroll algebra \(\mathfrak {ccarr}_{N}(d+1)\)  of level \(N\)  [6]}, [7]}, [8]}. It was shown that for \(N=2\)  (\(z=1\) ), the conformal Carroll algebra \(\mathfrak {ccarr}_{2}(d+1)\)  is isomorphic to BMS algebra in \(d+2\)  dimensions i.e. \(\mathfrak {ccarr}_{2}(d+1) \simeq \mathfrak {bms}_{(d+2)}\) . In particular, for our case of \(d=1\) , we have \(\mathfrak {ccarr}_{2}(2) \simeq \mathfrak {bms}_3\) . Also for Carrollian spacetimes describing null hypersurfaces embedded in pseudo-Riemannian spacetimes and for those obtained by the ultrarelativistic limit, the Weyl transformation has \(z=1\)  [1]}, [2]}.
